A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 3 - 2023 - Paper 1
Question 3
A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram.
Which of the following complex numbers is equal to $ar{z}$ ?
A. $-z$
B... show full transcript
Worked Solution & Example Answer:A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 3 - 2023 - Paper 1
Step 1
Determine $ar{z}$
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Answer
Given that z lies on the unit circle, we know that the conjugate of z, denoted as ar{z}, is represented as follows:
ar{z} = e^{-i heta}
where heta is the angle corresponding to z. This is due to the property of complex numbers on the unit circle.
Step 2
Identify $z$ based on given angle
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Answer
From the diagram, it can be observed that the angle corresponding to z is rac{ heta}{3}, leading to:
z = e^{i rac{ heta}{3}}
Step 3
Consider $z^2$
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